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| Spectral element method with a domain decomposition Stokes solver for steady cavity driven flow |
| Revised:October 23, 2004 |
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| DOI：10.7511/jslx20066125 |
| KeyWord:spectral element method,Stokes algorithm,Newton-Krylov method,cavity driven flow |
| MA Dong-jun LIU Yang SUN De-jun YIN Xie-yuan |
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| Abstract: |
| A Jacobian-Free-Newton-Krylov(JFNK) method with a time-stepping preconditioning technique is presented for the steady incompressible Navier-Stokes equations.The JFNK method combines the Newton method for superlinearly convergent solution of nonlinear equations and Krylov subspace method(such as GMRES) for solving the Newton correction equations.One crucial point for the JFNK method is constructing an effective preconditioner to reduce the number of Krylov iterations.The high order spectral element method with a domain decomposition Stokes solver is introduced as the effective preconditioner for the Newton iteration without forming the Jacobian matrix,which reduces the memory allocation,improves the poor conditioned system,speeds up the convergence rate.This algorithm is more suitable for the preconditioning than the usual time-splitting method and influence-matrix method,because it has no time-splitting divergence error and can be easily extended to the multi-dimensional flows.The numerical result for Kovasznay flow with an analytic solution shows the spectral accuracy with exponentially spatial convergence and superlinear convergence for Newton method.The method is applied to the two-dimensional constant velocity lid-driven problem at Reynolds number 100,400 and 1000.The results are compared with the benchmark data and show excellent agreement.An antisymmetric sinusoidal velocity driven cavity problem is considered for Re=800.Besides the stable patterns of steady symmetric and steady asymmetric solutions,a new pair of unsteady asymmetric solutions are found depending on the different initial conditions.This subcritical flow with multi-state can be served as an effective benchmark for the numerical solution of steady incompressible flows. |
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